Properties

Label 2.152.6t5.a.b
Dimension $2$
Group $S_3\times C_3$
Conductor $152$
Root number not computed
Indicator $0$

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Basic invariants

Dimension: $2$
Group: $S_3\times C_3$
Conductor: \(152\)\(\medspace = 2^{3} \cdot 19 \)
Artin stem field: 6.0.184832.1
Galois orbit size: $2$
Smallest permutation container: $S_3\times C_3$
Parity: odd
Determinant: 1.152.6t1.c.b
Projective image: $S_3$
Projective stem field: 3.1.2888.1

Defining polynomial

$f(x)$$=$\(x^{6} - 2 x^{5} + x^{4} + 2 x^{3} - 4 x + 3\)  Toggle raw display.

The roots of $f$ are computed in an extension of $\Q_{ 7 }$ to precision 7.

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 7 }$: \(x^{2} + 6 x + 3\)  Toggle raw display

Roots:
$r_{ 1 }$ $=$ \( 5 a + 4 + 4\cdot 7 + 2 a\cdot 7^{2} + \left(4 a + 6\right)\cdot 7^{3} + \left(a + 1\right)\cdot 7^{4} + \left(5 a + 5\right)\cdot 7^{5} + \left(6 a + 5\right)\cdot 7^{6} +O(7^{7})\)  Toggle raw display
$r_{ 2 }$ $=$ \( 2 a + 2 + 6 a\cdot 7 + \left(4 a + 2\right)\cdot 7^{2} + \left(2 a + 1\right)\cdot 7^{3} + \left(5 a + 6\right)\cdot 7^{4} + \left(a + 1\right)\cdot 7^{5} +O(7^{7})\)  Toggle raw display
$r_{ 3 }$ $=$ \( 6 a + \left(3 a + 1\right)\cdot 7 + 4\cdot 7^{2} + \left(a + 3\right)\cdot 7^{3} + \left(3 a + 2\right)\cdot 7^{4} + \left(3 a + 6\right)\cdot 7^{5} + 7^{6} +O(7^{7})\)  Toggle raw display
$r_{ 4 }$ $=$ \( a + 6 + \left(3 a + 5\right)\cdot 7 + 6 a\cdot 7^{2} + \left(5 a + 4\right)\cdot 7^{3} + \left(3 a + 4\right)\cdot 7^{4} + \left(3 a + 6\right)\cdot 7^{5} + \left(6 a + 5\right)\cdot 7^{6} +O(7^{7})\)  Toggle raw display
$r_{ 5 }$ $=$ \( 4 a + \left(4 a + 1\right)\cdot 7 + \left(2 a + 4\right)\cdot 7^{2} + \left(5 a + 1\right)\cdot 7^{3} + \left(4 a + 3\right)\cdot 7^{4} + \left(a + 5\right)\cdot 7^{5} +O(7^{7})\)  Toggle raw display
$r_{ 6 }$ $=$ \( 3 a + 4 + \left(2 a + 1\right)\cdot 7 + \left(4 a + 2\right)\cdot 7^{2} + \left(a + 4\right)\cdot 7^{3} + \left(2 a + 2\right)\cdot 7^{4} + \left(5 a + 2\right)\cdot 7^{5} + \left(6 a + 6\right)\cdot 7^{6} +O(7^{7})\)  Toggle raw display

Generators of the action on the roots $r_1, \ldots, r_{ 6 }$

Cycle notation
$(2,5,4)$
$(1,6,3)$
$(1,4)(2,6)(3,5)$

Character values on conjugacy classes

SizeOrderAction on $r_1, \ldots, r_{ 6 }$ Character value
$1$$1$$()$$2$
$3$$2$$(1,4)(2,6)(3,5)$$0$
$1$$3$$(1,6,3)(2,5,4)$$-2 \zeta_{3} - 2$
$1$$3$$(1,3,6)(2,4,5)$$2 \zeta_{3}$
$2$$3$$(1,6,3)$$-\zeta_{3}$
$2$$3$$(1,3,6)$$\zeta_{3} + 1$
$2$$3$$(1,3,6)(2,5,4)$$-1$
$3$$6$$(1,2,6,5,3,4)$$0$
$3$$6$$(1,4,3,5,6,2)$$0$

The blue line marks the conjugacy class containing complex conjugation.